3.661 \(\int \sqrt [3]{x} (a+b x)^2 \, dx\)

Optimal. Leaf size=36 \[ \frac{3}{4} a^2 x^{4/3}+\frac{6}{7} a b x^{7/3}+\frac{3}{10} b^2 x^{10/3} \]

[Out]

(3*a^2*x^(4/3))/4 + (6*a*b*x^(7/3))/7 + (3*b^2*x^(10/3))/10

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Rubi [A]  time = 0.0218017, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{3}{4} a^2 x^{4/3}+\frac{6}{7} a b x^{7/3}+\frac{3}{10} b^2 x^{10/3} \]

Antiderivative was successfully verified.

[In]  Int[x^(1/3)*(a + b*x)^2,x]

[Out]

(3*a^2*x^(4/3))/4 + (6*a*b*x^(7/3))/7 + (3*b^2*x^(10/3))/10

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Rubi in Sympy [A]  time = 4.04021, size = 34, normalized size = 0.94 \[ \frac{3 a^{2} x^{\frac{4}{3}}}{4} + \frac{6 a b x^{\frac{7}{3}}}{7} + \frac{3 b^{2} x^{\frac{10}{3}}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(1/3)*(b*x+a)**2,x)

[Out]

3*a**2*x**(4/3)/4 + 6*a*b*x**(7/3)/7 + 3*b**2*x**(10/3)/10

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Mathematica [A]  time = 0.0093291, size = 28, normalized size = 0.78 \[ \frac{3}{140} x^{4/3} \left (35 a^2+40 a b x+14 b^2 x^2\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^(1/3)*(a + b*x)^2,x]

[Out]

(3*x^(4/3)*(35*a^2 + 40*a*b*x + 14*b^2*x^2))/140

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Maple [A]  time = 0.005, size = 25, normalized size = 0.7 \[{\frac{42\,{b}^{2}{x}^{2}+120\,abx+105\,{a}^{2}}{140}{x}^{{\frac{4}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(1/3)*(b*x+a)^2,x)

[Out]

3/140*x^(4/3)*(14*b^2*x^2+40*a*b*x+35*a^2)

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Maxima [A]  time = 1.34542, size = 32, normalized size = 0.89 \[ \frac{3}{10} \, b^{2} x^{\frac{10}{3}} + \frac{6}{7} \, a b x^{\frac{7}{3}} + \frac{3}{4} \, a^{2} x^{\frac{4}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^2*x^(1/3),x, algorithm="maxima")

[Out]

3/10*b^2*x^(10/3) + 6/7*a*b*x^(7/3) + 3/4*a^2*x^(4/3)

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Fricas [A]  time = 0.204386, size = 36, normalized size = 1. \[ \frac{3}{140} \,{\left (14 \, b^{2} x^{3} + 40 \, a b x^{2} + 35 \, a^{2} x\right )} x^{\frac{1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^2*x^(1/3),x, algorithm="fricas")

[Out]

3/140*(14*b^2*x^3 + 40*a*b*x^2 + 35*a^2*x)*x^(1/3)

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Sympy [A]  time = 6.90639, size = 1953, normalized size = 54.25 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(1/3)*(b*x+a)**2,x)

[Out]

Piecewise((27*a**(34/3)*(-1 + b*(a/b + x)/a)**(1/3)/(-140*a**8*b**(4/3) + 420*a*
*7*b**(7/3)*(a/b + x) - 420*a**6*b**(10/3)*(a/b + x)**2 + 140*a**5*b**(13/3)*(a/
b + x)**3) + 27*a**(34/3)*exp(16*I*pi/3)/(-140*a**8*b**(4/3) + 420*a**7*b**(7/3)
*(a/b + x) - 420*a**6*b**(10/3)*(a/b + x)**2 + 140*a**5*b**(13/3)*(a/b + x)**3)
- 72*a**(31/3)*b*(-1 + b*(a/b + x)/a)**(1/3)*(a/b + x)/(-140*a**8*b**(4/3) + 420
*a**7*b**(7/3)*(a/b + x) - 420*a**6*b**(10/3)*(a/b + x)**2 + 140*a**5*b**(13/3)*
(a/b + x)**3) - 81*a**(31/3)*b*(a/b + x)*exp(16*I*pi/3)/(-140*a**8*b**(4/3) + 42
0*a**7*b**(7/3)*(a/b + x) - 420*a**6*b**(10/3)*(a/b + x)**2 + 140*a**5*b**(13/3)
*(a/b + x)**3) + 60*a**(28/3)*b**2*(-1 + b*(a/b + x)/a)**(1/3)*(a/b + x)**2/(-14
0*a**8*b**(4/3) + 420*a**7*b**(7/3)*(a/b + x) - 420*a**6*b**(10/3)*(a/b + x)**2
+ 140*a**5*b**(13/3)*(a/b + x)**3) + 81*a**(28/3)*b**2*(a/b + x)**2*exp(16*I*pi/
3)/(-140*a**8*b**(4/3) + 420*a**7*b**(7/3)*(a/b + x) - 420*a**6*b**(10/3)*(a/b +
x)**2 + 140*a**5*b**(13/3)*(a/b + x)**3) - 60*a**(25/3)*b**3*(-1 + b*(a/b + x)/
a)**(1/3)*(a/b + x)**3/(-140*a**8*b**(4/3) + 420*a**7*b**(7/3)*(a/b + x) - 420*a
**6*b**(10/3)*(a/b + x)**2 + 140*a**5*b**(13/3)*(a/b + x)**3) - 27*a**(25/3)*b**
3*(a/b + x)**3*exp(16*I*pi/3)/(-140*a**8*b**(4/3) + 420*a**7*b**(7/3)*(a/b + x)
- 420*a**6*b**(10/3)*(a/b + x)**2 + 140*a**5*b**(13/3)*(a/b + x)**3) + 135*a**(2
2/3)*b**4*(-1 + b*(a/b + x)/a)**(1/3)*(a/b + x)**4/(-140*a**8*b**(4/3) + 420*a**
7*b**(7/3)*(a/b + x) - 420*a**6*b**(10/3)*(a/b + x)**2 + 140*a**5*b**(13/3)*(a/b
+ x)**3) - 132*a**(19/3)*b**5*(-1 + b*(a/b + x)/a)**(1/3)*(a/b + x)**5/(-140*a*
*8*b**(4/3) + 420*a**7*b**(7/3)*(a/b + x) - 420*a**6*b**(10/3)*(a/b + x)**2 + 14
0*a**5*b**(13/3)*(a/b + x)**3) + 42*a**(16/3)*b**6*(-1 + b*(a/b + x)/a)**(1/3)*(
a/b + x)**6/(-140*a**8*b**(4/3) + 420*a**7*b**(7/3)*(a/b + x) - 420*a**6*b**(10/
3)*(a/b + x)**2 + 140*a**5*b**(13/3)*(a/b + x)**3), Abs(b*(a/b + x)/a) > 1), (-2
7*a**(34/3)*(1 - b*(a/b + x)/a)**(1/3)*exp(16*I*pi/3)/(-140*a**8*b**(4/3) + 420*
a**7*b**(7/3)*(a/b + x) - 420*a**6*b**(10/3)*(a/b + x)**2 + 140*a**5*b**(13/3)*(
a/b + x)**3) + 27*a**(34/3)*exp(16*I*pi/3)/(-140*a**8*b**(4/3) + 420*a**7*b**(7/
3)*(a/b + x) - 420*a**6*b**(10/3)*(a/b + x)**2 + 140*a**5*b**(13/3)*(a/b + x)**3
) + 72*a**(31/3)*b*(1 - b*(a/b + x)/a)**(1/3)*(a/b + x)*exp(16*I*pi/3)/(-140*a**
8*b**(4/3) + 420*a**7*b**(7/3)*(a/b + x) - 420*a**6*b**(10/3)*(a/b + x)**2 + 140
*a**5*b**(13/3)*(a/b + x)**3) - 81*a**(31/3)*b*(a/b + x)*exp(16*I*pi/3)/(-140*a*
*8*b**(4/3) + 420*a**7*b**(7/3)*(a/b + x) - 420*a**6*b**(10/3)*(a/b + x)**2 + 14
0*a**5*b**(13/3)*(a/b + x)**3) - 60*a**(28/3)*b**2*(1 - b*(a/b + x)/a)**(1/3)*(a
/b + x)**2*exp(16*I*pi/3)/(-140*a**8*b**(4/3) + 420*a**7*b**(7/3)*(a/b + x) - 42
0*a**6*b**(10/3)*(a/b + x)**2 + 140*a**5*b**(13/3)*(a/b + x)**3) + 81*a**(28/3)*
b**2*(a/b + x)**2*exp(16*I*pi/3)/(-140*a**8*b**(4/3) + 420*a**7*b**(7/3)*(a/b +
x) - 420*a**6*b**(10/3)*(a/b + x)**2 + 140*a**5*b**(13/3)*(a/b + x)**3) + 60*a**
(25/3)*b**3*(1 - b*(a/b + x)/a)**(1/3)*(a/b + x)**3*exp(16*I*pi/3)/(-140*a**8*b*
*(4/3) + 420*a**7*b**(7/3)*(a/b + x) - 420*a**6*b**(10/3)*(a/b + x)**2 + 140*a**
5*b**(13/3)*(a/b + x)**3) - 27*a**(25/3)*b**3*(a/b + x)**3*exp(16*I*pi/3)/(-140*
a**8*b**(4/3) + 420*a**7*b**(7/3)*(a/b + x) - 420*a**6*b**(10/3)*(a/b + x)**2 +
140*a**5*b**(13/3)*(a/b + x)**3) - 135*a**(22/3)*b**4*(1 - b*(a/b + x)/a)**(1/3)
*(a/b + x)**4*exp(16*I*pi/3)/(-140*a**8*b**(4/3) + 420*a**7*b**(7/3)*(a/b + x) -
420*a**6*b**(10/3)*(a/b + x)**2 + 140*a**5*b**(13/3)*(a/b + x)**3) + 132*a**(19
/3)*b**5*(1 - b*(a/b + x)/a)**(1/3)*(a/b + x)**5*exp(16*I*pi/3)/(-140*a**8*b**(4
/3) + 420*a**7*b**(7/3)*(a/b + x) - 420*a**6*b**(10/3)*(a/b + x)**2 + 140*a**5*b
**(13/3)*(a/b + x)**3) - 42*a**(16/3)*b**6*(1 - b*(a/b + x)/a)**(1/3)*(a/b + x)*
*6*exp(16*I*pi/3)/(-140*a**8*b**(4/3) + 420*a**7*b**(7/3)*(a/b + x) - 420*a**6*b
**(10/3)*(a/b + x)**2 + 140*a**5*b**(13/3)*(a/b + x)**3), True))

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GIAC/XCAS [A]  time = 0.204426, size = 32, normalized size = 0.89 \[ \frac{3}{10} \, b^{2} x^{\frac{10}{3}} + \frac{6}{7} \, a b x^{\frac{7}{3}} + \frac{3}{4} \, a^{2} x^{\frac{4}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^2*x^(1/3),x, algorithm="giac")

[Out]

3/10*b^2*x^(10/3) + 6/7*a*b*x^(7/3) + 3/4*a^2*x^(4/3)